Dear dagn, 
 
eng789 is correct.
 
Oxford Dictionary of English
 
"Draw blood" = "Cause someone to bleed, especially in the course of a fight".
Example: "The blow drew blood from the corner of his mouth".
 
The writer has long fingernails and could have written this:
 
"I grip my neck, like a bird with long claws, and I begin to claw my neck so hard that my fingernails enter my skin.  Blood begins to come from the deep scratch, the wound.
I draw blood. I make myself bleed by scratching my neck very hard".
 
I hope that this helps you.
 
Les